Management of a Shared Spectrum Network inWireless Communications

Shining Wu∗, Jiheng Zhang†, Rachel Q. Zhang†∗Department of Logistics and Maritime Studies

The Hong Kong Polytechnic University, Hong Kong†Department of Industrial Engineering and Logistics ManagementThe Hong Kong University of Science and Technology, Hong Kong

sn.wu@polyu.edu.hk, jiheng@ust.hk, rzhang@ust.hk

We consider a band of the electromagnetic spectrum with a finite number of identical channels shared

by both licensed and unlicensed users. Such a network differs from most many-server, two-class queues

in service systems including call centers due to the restrictions imposed on the unlicensed users in order

to limit interference to the licensed users. We first approximate the key performance indicators, namely

the throughput rate of the system and the delay probability of the licensed users under the asymptotic

regime, which requires the analysis of both scaled and unscaled processes simultaneously using the averaging

principle. Our analysis reveals a number of distinctive properties of the system. For example, sharing does

not affect the level of service provided to the licensed users in an asymptotic sense even when the system

is critically loaded. We then study the optimal sharing decisions of the system to maximize the system

throughput rate while maintaining the delay probability of the licensed users below a certain level when the

system is overloaded. Finally, we extend our study to systems with time-varying arrival rates and propose a

diffusion approximation to complement our fluid one.

Key words : spectrum management; many-server queues; fluid approximation; averaging principle

History : Manuscript OPRE-2015-02-088

1. Introduction

The radio spectrum refers to the range of frequencies suitable for wireless communications in

television and radio broadcasting, aviation, public safety, cell phones, and so on. Until recently,

spectrum regulatory bodies including the Federal Communications Commission (FCC) in the US

and the European Telecommunications Standards Institute (ETSI) have always allocated spectrum

bands exclusively to certain service providers whose users are referred to as primary or licensed

users, often based on the radio technologies available at the time of allocation. Such static spectrum

allocation mitigates interference to essential services, yet it creates underutilization of the allocated

spectrum, which can be below 20% even during high demand periods in certain geographic areas.

For instance, during the high demand period of a political convention held in New York City in

2004, only about 13% of the allocated spectrum was utilized (Prasad et al. 2010). Studies conducted

by the FCC, universities, and industry also revealed that a major part of the spectrum is not fully

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utilized most of the time. On the other hand, over the past decades, the convergence of voice and

data in wireless communications triggered by the convergence of wireless and Internet technologies

has led to an explosion in the number of bits transmitted over the air (Biglieri et al. 2013). Since

it is usually difficult to open up higher frequency bands for mobile applications as transmission

becomes less reliable in those bands, the existing radio spectrum for data transmission is reaching

its capacity.

A natural approach to alleviate the artificial scarcity of spectrum due to static allocation is to

allow opportunistic use of temporarily idle channels by unlicensed or secondary users to increase

the throughput of already allocated spectrum. This is referred to as opportunistic spectrum access

(Hossain et al. 2009). However, allowing unlicensed users access may cause interference to existing

licensed users. Thus, such paradigm of operation requires (1) the knowledge of the state of frequency

bands (e.g., channel availability, queues, etc.) in real time and (2) an effective control mechanism

to govern spectrum usage by unlicensed users, which led to the development of the concept of

cognitive radio, first introduced by Mitola and Maguire (1999). Using advanced radio and signal

processing technology, cognitive radio is a software-defined radio device that can intelligently sense

and explore the spectrum environment, track changes, communicate information among different

transceivers and react according to a control mechanism (Hossain et al. 2009). It is widely regarded

as one of the most promising technologies for future wireless communications and may potentially

mitigate, through dynamic spectrum access, the problem of radio spectrum scarcity.

It is obvious that implementation of a cognitive radio network involves both technological and

operational issues, yet much of the research is focused on the former (see Section 2.1 for some

relevant literature). In this paper, we focus on the operational issues by considering a band of

spectrum with multiple identical channels shared by both licensed and unlicensed users. Since the

spectrum has already been allocated to the licensed users and it is usually difficult to set aside

a subset of channels for either groups in reality for technical reasons, we assume all the channels

are accessible by both licensed and unlicensed users as in most existing literature in electrical

engineering. Furthermore, although concurrent transmission is allowed in some networks under

which the main concern is technological (e.g., the power level at which an unlicensed user is allowed

to transmit), we focus on systems where each channel serves only one user at a time, referred

to as the interweave paradigm (Biglieri et al. 2013). Thus, the network considered is a two-class

queue served by a single pool of homogeneous servers as in applications in service systems such as

call centers and healthcare but with some distinctive features due to the restrictions imposed on

the unlicensed users (Hossain et al. 2009). (1) When all the channels are occupied upon arrival, a

licensed user will join a queue along with other waiting licensed users who will be served first-in-

first-out (FIFO) as soon as a channel becomes available, while an unlicensed user will join a queue

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along with other waiting unlicensed users and will only be allowed to sense channel availability

periodically. An unlicensed user can only occupy a channel when an available channel is detected

and no licensed users are waiting, and may also abandon the system every time he senses but finds

no available channel. Such a queue where users wait for retrial is referred to as an orbit queue

in the queueing literature and is common in computer and communications networks (Artalejo

and Gomez-Corral 2008). (2) When in transmission, a licensed user can transmit until his service

requirement is fulfilled, while an unlicensed user is only allocated a fixed amount of time, referred

to as a service session, approaching the end of which he has to stop transmission to sense the

environment as sensing cannot occur simultaneously with data transmission. He will be allowed

to continue for another service session only if he senses no waiting licensed users. Otherwise, he

has to release the channel and join the orbit queue along with other unlicensed users or abandon

the system if he needs more time. Note that data transmission can be interrupted and resumed,

hence more complicated control policies than those in call centers are allowed, which leads to new

managerial insights.

Assuming that perfect sensing can be achieved in a fixed amount of time and both licensed and

unlicensed users arrive according to Poisson processes, we first perform in-depth analysis on the

key performance indicators in the management of shared spectrum networks, namely the delay

probability of the licensed users and the system throughput rate. We then focus on the restrictions

that need to be imposed on the unlicensed users when in service and waiting, i.e., the length of

a service session and the sensing frequency while waiting. Intuitively, the longer a service session

is, the less sensing an unlicensed user needs to perform and hence a higher system throughput

rate. Yet, longer service sessions can cause more interference to the licensed users. Likewise, the

more frequently an unlicensed user senses channel availability while waiting, the sooner he is able

to find an available channel but the more interference he causes to the licensed users. Thus, there

is a tradeoff between the throughput rate and the level of interference to the licensed users when

deciding on the length of a service session and the sensing frequency. The goals of this research are

to answer the following questions: (1) Should a given band of spectrum be shared with unlicensed

users? (2) When sharing is permitted, how long should unlicensed users be allowed to transmit each

time they occupy a channel and how frequently should they be allowed to sense channel availability

while waiting? (3) Under what conditions is sharing more beneficial? (4) How will the decision

change with uncertain arrival rates or time-varying arrivals?

Since the band of spectrum considered usually consists of hundreds or thousands of channels, we

can treat the system as a large network, and approximate the performance under the asymptotic

regime as in Gupta and Kumar (2000) and El Gamal et al. (2006). Due to the restrictions imposed

on the unlicensed users when in service and waiting, we need to analyze both scaled and unscaled

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processes simultaneously using the averaging principle, i.e., approximating the unscaled process

by its long-run average. We then formulate the problem as finding the optimal restrictions on

the unlicensed users to maximize the throughput rate while maintaining the delay probability of

licensed users below a certain level. Our main findings are as follows:

1. Sensing frequency of the unlicensed users while waiting: Surprisingly, sensing fre-

quency does not affect the system performance asymptotically as long as the unlicensed users

are required to sense channel availability, which takes time and prevents them from occupying

idle channels instantaneously. Thus, there is no need to impose any restriction on the sensing

frequency from the operational perspective. The decision thus should primarily be based on

technological concerns, for instance, power consumption associated with each sensing activity.

2. The length of a service session: Intuitively, shorter service sessions should cause less

interference to and hence lower the delay probability of the licensed users. However, with

shorter service sessions, the unlicensed users need more service sessions to finish their service

and hence need to perform more sensing activities while occupying a channel. Thus, shorter

service sessions do not always improve the delay probability.

3. Optimal sharing decisions: When the system is under or critically loaded, the interference

of the unlicensed users to the licensed users is negligible and there is no need to impose a

restriction on the service process of the unlicensed users either. That is, allowing the unlicensed

users to complete their transmissions without restriction will not cause any interference to

licensed users asymptotically as the delay probability is 0. This result is very different from

that of most non-preemptive queueing systems under which the delay probability is strictly

between 0 and 1 when the system is critically loaded.

When the system is overloaded, the delay probability of the licensed users is quasi-convex

in the length of the service sessions of the unlicensed users, strictly between 0 and 1 and

increasing in the load. Thus, a restriction on the service process of the unlicensed users should

be imposed only when the load is above a threshold. Furthermore, a shorter service session

should be allocated as the load increases until spectrum sharing is no longer feasible.

The insight that it is possible to improve spectrum utilization while guaranteeing a very high

service level, expected by licensed users in practice, is very encouraging news. Thus, spectrum

sharing can potentially be a socially optimal solution to alleviating spectrum scarcity.

4. For a given system load, a shorter service session should be allocated to the unlicensed users

(1) as the proportion of the licensed users increases, (2) if there are fewer licensed users with

longer service times, or (3) if there are more unlicensed users with shorter service times. As

the service session shortens, more unlicensed users will abandon the system, which lowers

the throughput rate under scenarios (1) and (2). Therefore, spectrum sharing is beneficial to

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systems with a smaller proportion of licensed users or a large number of licensed users with

shorter service times.

5. When the arrival rates are time varying, a shorter service session should be allocated to the

unlicensed users during busy periods. Although optimal control requires continuous adjust-

ment in real time, near-optimal control can be accomplished with occasional adjustments.

To the best of our knowledge, this is the first comprehensive study of a shared network in wireless

communications. Although there have been some attempts by researchers in electrical engineering

using relatively simple queueing models, our model captures many more of the features of such

a system. We are able to uncover complicated system dynamics and obtain managerial insights

different from those drawn from the many well-studied service systems. Our work not only opens

the door for new applications of existing queueing theory in wireless communications, but may also

stimulate the development of new methodologies.

The remainder of this paper is organized as follows. We review the relevant literature in both

electrical engineering and queueing theory in the next section and describe the problem of dynamic

spectrum sharing in detail in Section 3. In Section 4, we provide a fluid approximation and study the

optimal sharing decisions of the system. In Section 5, we offer the intuition behind the construction

of the fluid model and give justifications for the fluid approximation. We extend our analysis to

systems with time-varying arrival rates and discuss a diffusion scaled approximation in Section 6.

We conclude our paper and provide some future research directions in Section 7. The proofs can

all be found in the Appendix.

2. Literature Review

In this section, we will first provide some background on the research on opportunistic spectrum

access, mostly in electrical engineering. Since we will model a shared network as a multi-class,

many-server queue where the unlicensed users join an orbit queue and analyze it using the averaging

principle, we will review the relevant literature in queueing theory and its applications.

2.1. On Opportunistic Spectrum Access

Most of the work on opportunistic spectrum access focuses on the technological issues such as the

sensing technology to detect idle channels (Mishra et al. 2006), signal encoding (Devroye et al.

2006) and the control of the transmit power to limit interference (Bansal et al. 2008). For research

on various technological issues associated with cognitive radio, readers may refer to Akyildiz et al.

(2006) and Goldsmith et al. (2009).

Research on the operational issues under simplified settings, however, remains scant. Huang

et al. (2008) perform an analytical study on a single-channel system with one licensed and one

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unlicensed user, as well as numerical studies on a multi-channel system. They also consider the

decisions on the sensing frequency of unlicensed users and how long unlicensed users should be

allowed to transmit in their numerical study. Zhao et al. (2008) study the optimal access strategy

of an unlicensed user based on the sensing outcome given that each channel has already been

assigned to a specific licensed user, while Capar et al. (2002) compare the system performance in

terms of bandwidth utilization and blocking probability when a licensed user can be assigned to

any channel randomly or in a controlled way.

For a more comprehensive picture of the various issues in dynamic spectrum management and

cognitive radio networks, readers may refer to Hossain et al. (2009) and Biglieri et al. (2013).

2.2. On Queueing Theory and Applications

Multi-Class, Many-Server Queues Since a band of spectrum consists of hundreds or thou-

sands of channels and there are both licensed and unlicensed users, the literature of multi-class,

many-server queues is relevant. The study of many-server queues was substantiated by the seminal

work of Halfin and Whitt (1981), who derive the steady-state distribution of the diffusion limits

and establish the square root law describing the relationship between the system load and delay

probability. The mathematical insights of the square root law have since been extended and widely

adopted in the daily management of call centers around the world. Later, Puhalskii and Reiman

(2000) extend the study to multi-class models.

There is a large body of work on multi-class, many-server systems due to their applications

in call centers, manufacturing and computer-communication systems with a focus on asymptotic

optimal control of the underlying systems. For example, Atar et al. (2004) study asymptotic optimal

schedule policies, Gurvich and Whitt (2009) propose a family of queue-and-idleness-ratio rules for

routing and scheduling, and Maglaras and Zeevi (2004, 2005) examine the pricing, capacity sizing

and admission control decisions in a differentiated service system with guaranteed (high priority)

and best-effort (low priority) users. Our model differs from the existing work in that the service

(i.e., data transmission) of the unlicensed users may be fulfilled after multiple interruptions, which

is not the case in most other applications.

Since the service of unlicensed users may be interrupted by waiting licensed users, the literature

on queues with service interruption caused by preemptive priority, which dates back to White and

Christie (1958) in single server settings, is also relevant. For a review on some of the early work,

we refer the reader to Jaiswal (1968). Among the existing work, most focuses on characterizing

the steady state distributions of the queue length, the sojourn time and so on for a given priority

discipline. For example, Brosh (1969) derives the expressions for the expected time from arrival

to inception of service and provides bounds for the expected sojourn time for each class when

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all classes have the same service rates. Buzen and Bondi (1983) obtain the exact expressions for

the mean sojourn times when all classes have the same service rates and provide approximations

when different classes have different service rates. Recently, Wang et al. (2015) conduct the exact

analysis of the steady state of a preemptive M/M/c queue when different classes have different

service rates. In our paper, we focus on the control of the service process of the unlicensed users,

i.e., how their service processes should be interrupted.

Orbit Queues Since the unlicensed users join an orbit queue in our setting, the literature along

this line is also relevant. Yang and Templeton (1987) and Falin and Templeton (1997) offer a survey

and a comprehensive summary of the earlier papers, respectively. Later, Mandelbaum et al. (2002)

provide an analytical approximation to the key performance of a many-server queueing system with

abandonment and retrials under an asymptotic regime. In all these papers, even though customers

may join an orbit queue for retrial if they cannot be served immediately upon arrival, their service

cannot be interrupted once started.

Recently, a number of studies consider systems where customers may require repeat service due

to unresolved or new issues. For instance, de Vericourt and Zhou (2005) and Zhan and Ward (2014)

study a customer-routing problem in call centers with callbacks, while de Vericourt and Jennings

(2008) and Yom-Tov and Mandelbaum (2014) examine a staffing problem for membership services

and healthcare systems where customers may require multiple rounds of service. These systems

differ from ours in that customers will wait in a FIFO queue for retrial if the systems are busy upon

arrival, although they will first join an orbit queue after they have had a round of service. Allowing

the unlicensed users to retry and join an orbit queue as in our setting significantly complicates the

analysis since there may be a large number of customers switching frequently between being in

service and being in the orbit queue.

The Averaging Principle Only a few studies in the queueing literature have required the

use of the averaging principle. Building on a fundamental theory of the averaging principle by

Kurtz (1992), Hunt and Kurtz (1994) study martingales and related random measures of large loss

networks. Whitt (2002) summarizes the early studies on scheduling multi-class queues using the

averaging principle. Recently, a series of studies by Perry and Whitt (2011a,b, 2013) has applied

the averaging principle to obtain both the fluid and diffusion limits for an overloaded X model of

many-server queues, and to derive insights about the asymptotic optimal control of the system.

Pang and Perry (2015) apply the averaging principle to obtain a logarithmic safety staffing rule

for call centers with call blending. We adopt some of the methodologies developed by Hunt and

Kurtz (1994) and Perry and Whitt (2011a).

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3. Problem Description and Assumptions

3.1. The Sharing Network and Performance Measures

We consider a band of spectrum consisting of n identical channels shared by both licensed and

unlicensed users, denoted as user types 1 and 2 respectively, and each channel can only be occupied

by one user at a time. That is, concurrent transmission is not allowed. Furthermore, we assume

that perfect sensing can be achieved in a fixed amount of time 1µs

by an unlicensed user. Type i

users arrive according to a Poisson process with the rate λni and require an exponential amount

of service time with the rate µi, i= 1,2. If there is an available channel, an arriving licensed user

will occupy it immediately until his service requirement is fulfilled. Otherwise, he will join a queue

along with other waiting licensed users who will be served FIFO as the channels become available.

Next, we describe the service and waiting processes of the unlicensed users in the shared network

illustrated in Figure 1 where In(t) is the number of idle channels and Qni (t) is the queue length of

type i users at time t. According to the policy,

Qn1 (t)In(t) = 0. (1)

Upon arrival, an unlicensed user will occupy a channel if there is one available. Otherwise, he will

join an orbit queue along with other waiting unlicensed users with probability 1− φ or abandon

the system.

• The service process: Once he occupies a channel, an unlicensed user is allocated a fixed amount

of uninterrupted time, referred to as a service session (Liu and Wang 2010), regardless of his

service requirement. If he needs more time and finds no licensed user waiting at the end of

a session through sensing, he is allowed to continue for another service session. Since sensing

cannot occur simultaneously with data transmission and must be interweaved, he needs to

devote the last 1µs

amount of time in each service session to sense the environment if he needs

more time. Hence, we denote the length of a service session by 1µt

+ 1µs

where 1µt

is the amount

of time allowed for transmission in a service session. If the unlicensed user completes his

transmission within 1µt

amount of time in a session, he will release the channel without sensing

and leave the system. Otherwise, he will have to sense the environment and his service will be

interrupted if he finds a waiting licensed user, in which case he will join the orbit queue with

probability 1−φ or abandon the system.

• The waiting process: While waiting in the orbit queue, an unlicensed user will only be allowed

to sense channel availability periodically. Let 1θ

denote the time between sensing activities,

which includes the time needed for sensing channel availability. After each sensing activity,

he will occupy a channel if he finds an idle one. Otherwise, he will abandon the system with

probability φ, or stay in the orbit queue for another sensing activity with probability 1−φ.

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As one can see, the network has its distinctive characteristics which are not present in most exist-

ing multi-class, many-server queueing systems due to the restrictions on the service and waiting

processes of the unlicensed users, i.e., the transmission time 1µt

in a service session and the sensing

frequency θ. The data transmission of the unlicensed users can be interrupted and resumed for

any number of times and sensing for channel availability by the unlicensed users in the queue is

only allowed periodically. As a result, an unlicensed user may abandon the system upon arrival,

after spending some time in the queue without receiving any service, or after receiving partial

service. Furthermore, each unlicensed user in the orbit queue needs to sense channel availability

independently, which guarantees certain idleness in the system even when there are waiting unli-

censed users. These features are new in the queueing literature and interesting, yet significantly

complicate the analysis.

n Channels

Wait & Sense

Queueλn1

λn2

In(t)> 0

In(t) = 0 1−φ

Abandon

φ

need moreservice

Qn1 (t) = 0

Qn1(t)

>01−φ

Abandon

φ

In(t)> 0

In(t)

=0

1−φ

Abandon

φ

- - - licensed usersunlicensed users

Figure 1 The spectrum sharing network

The performance measures we are concerned with are the throughput rate of the system and

the probability that all the channels are occupied upon the arrival of a licensed user, referred to

as the delay probability. The goal is to find the transmission time 1µt

in a service session and the

sensing frequency θ of the unlicensed users that maximize the throughput of the unlicensed users

while guaranteeing the delay probability of the licensed users below a certain level.

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3.2. Modeling Assumptions

Since the problem is analytically intractable, we will first approximate the deterministic transmis-

sion time, sensing time, and the time between consecutive sensing activities in the orbit queue

by the exponential distributions with the same means. Table 1 presents a simulation study of the

delay probability of the licensed users and the throughput rate of the unlicensed users with deter-

ministic times and exponential times when λ1 = 0.2, λ2 = 0.9, 1µ1

= 1µ2

= 1, 1µs

= 0.001, θ= 0.4, and

φ= 0.5. For 1µt∈ {∞,0.6,0.2}, we set n= 100,500,1000,2000,4000 and let λni = nλi. We report the

means and 0.95 confidence intervals of the delay probabilities and throughput rates. As one can

see, approximating the deterministic times by the exponential times does not reduce the accuracy

very much, especially when n is large as in our application where n is in the hundreds or thousands.

1µt

ndelay probability throughput rate

Deterministic Exponential Deterministic Exponential

∞

100 0.2528± .0036 0.2531± .0044 0.7682± .0014 0.7678± .0024500 0.2120± .0014 0.2142± .0019 0.7928± .0007 0.7915± .00121000 0.2075± .0017 0.2075± .0018 0.7957± .0008 0.7957± .00092000 0.2045± .0009 0.2044± .0008 0.7975± .0005 0.7974± .00054000 0.2014± .0010 0.2021± .0008 0.7992± .0004 0.7988± .0004Fluid 0.1995 0.8000

0.6

100 0.2360± .0040 0.2314± .0036 0.7656± .0021 0.7662± .0022500 0.1979± .0009 0.1953± .0011 0.7901± .0007 0.7899± .00051000 0.1906± .0013 0.1892± .0007 0.7949± .0006 0.7939± .00052000 0.1872± .0006 0.1854± .0010 0.7968± .0004 0.7964± .00054000 0.1854± .0003 0.1838± .0006 0.7979± .0002 0.7973± .0003Fluid 0.1813 0.7987

0.2

100 0.2262± .0031 0.2259± .0033 0.7643± .0023 0.7641± .0019500 0.1916± .0024 0.1918± .0020 0.7878± .0014 0.7871± .00111000 0.1860± .0010 0.1855± .0010 0.7914± .0005 0.7914± .00072000 0.1820± .0011 0.1820± .0010 0.7940± .0007 0.7938± .00064000 0.1802± .0005 0.1804± .0007 0.7954± .0003 0.7949± .0004Fluid 0.1784 0.7960

Table 1 Comparison of performance measures with deterministic vs. exponential times.

With the exponential times mentioned above, the probability that an unlicensed user will com-

plete his transmission in a service session is given by p= µ2µ2+µt

. Furthermore, the actual amount

of time an unlicensed user will occupy a channel in each service session follows a phase-type dis-

tribution with mean1

µ=

1

µ2 +µt+ (1− p) · 1

µs=

µt +µs(µ2 +µt)µs

, (2)

which is less than the allocated session time 1µt

+ 1µs

. If we let Zni (t) denote the number of channels

occupied by type i users at time t, the instantaneous throughput rate at time t is given by pµZn2 (t).

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With hundreds or thousands of channels in a band of spectrum, performing an analytical study

of the shared network under a large system scaling to be defined below is not only for technical

tractability, but also appropriate.

Definition 1 (Asymptotic Regime). There exist positive real numbers λi, i= 1,2, such that

limn→∞

λnin

= λi, andλ1

µ1

< 1.

Here λi represents the size of type i users. Different cognitive radio networks have different

proportions of licensed and unlicensed users. In IEEE 802.22 WRANs, unlicensed users outnumber

licensed users (Zhang et al. 2009, Jia et al. 2008), i.e., λ2 >λ1, while in TV white space networks,

licensed users are the majority (van de Beek et al. 2012), i.e., λ1 > λ2. In Gong et al. (2015), the

licensed users (from a down-link cellular system) and the unlicensed users (from an ad hoc network)

have comparable numbers, i.e., λ1 ≈ λ2.

Under the asymptotic regime, we will add a bar to the existing notation to represent the scaled

processes in our model, e.g., Qni (t) =

Qni (t)

n, and use the lower case, e.g., qi(t), to represent the

corresponding fluid model, which will be proven to be the fluid limit of the scaled processes.

4. Main Results and Insights

Under the asymptotic regime, the processes involved are scaled and then approximated by tractable

ones that preserve the relevant information about the system performance. As in most multi-class

queueing systems, the queue length of the licensed users, who have a higher priority, will vanish

asymptotically. This is not a problem if the queue length of the licensed users does not affect the

users in service in an asymptotic sense, which is the case in most applications, and one can still

obtain the managerial insights by analyzing the limit of scaled processes alone. However, whether

the number of waiting license users is asymptotically small or exactly zero is important in our

setting as it determines whether an unlicensed user should vacate a channel but the scaled processes

fail to preserve such important information. Thus, the analysis requires information from both

scaled and unscaled processes, involves tracking the two processes simultaneously, and needs to use

the averaging principle. These requirements are rare in the literature with only a few exceptions

such as Perry and Whitt (2011a), Luo and Zhang (2013), Pang and Perry (2015).

In this section, we first introduce our fluid model x(t) = (z1(t), q1(t), z2(t), q2(t)) which will be

used to approximate the stochastic process Xn(t) = (Zn1 (t),Qn1 (t),Zn2 (t),Qn

2 (t)) in our system with

the justifications to be provided in Section 5. We then derive the steady-state performance and

study the optimal sharing decisions of the system in the steady state using the fluid approximations.

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4.1. The Fluid Model

Definition 2 (FLUID MODEL). The process x(t) = (z1(t), q1(t), z2(t), q2(t)) evolves according

to the constraint

0 = [1− z1(t)− z2(t)]q1(t), (3)

and the following differential equations

z′1(t) = [1−β(t)]λ1 +α(t)[µ1z1(t) +µz2(t)]−µ1z1(t), (4)

q′1(t) = β(t)λ1−α(t)[µ1z1(t) +µz2(t)], (5)

z′2(t) = [1−β(t)][λ2 + θq2(t)]− [p+α(t)(1− p)]µz2(t), (6)

q′2(t) = (1−φ)β(t)[λ2 + θq2(t)] + (1−φ)α(t)(1− p)µz2(t)− θq2(t), (7)

where β(t) and α(t) depend on how constraint (3) is met. If q1(t) > 0, then β(t) = α(t) = 1; if

z1(t) + z2(t)< 1, then β(t) = α(t) = 0; otherwise,

β(t) = min

{([λ1 +λ2 + θq2(t)−µ1z1(t)− pµz2(t)][µ1z1(t) +µz2(t)]

[λ1 +λ2 + θq2(t)][µ1z1(t) +µz2(t)]−λ1[µ1z1(t) + pµz2(t)]

)+

,1

}, (8)

α(t) = min

{λ1β(t)

µ1z1(t) +µz2(t),1

}. (9)

The fluid model defined above is built on the evolution of the system described in Section 3. As

we will explain in Section 5 and define formally in Appendix B, β(t) is the instantaneous delay

probability of the licensed users and α(t) is the instantaneous probability that an unlicensed user

has to release the channel after a service session (i.e., there are waiting licensed users in the system),

referred to as the interruption probability, under the fluid model. Thus, the differential equations

(4)–(7) are quite intuitive. Take equation (4) for an example. The rate of increase in z1(t) consists

of two parts: (1) When the licensed users arrive (at the rate λ1), there is an available channel

(with probability 1− β(t)); (2) When the licensed users finish service (at the rate µ1z1(t)) or the

unlicensed users finish a service session (at the rate µz2(t)); there are waiting licensed users (with

probability α(t)). The rate of decrease in z1(t) is µ1z1(t), which is the rate the licensed users

occupying the channels finish service. For equation (7), the rate of increase in q2(t) consists of two

parts: (1) When the unlicensed users arrive or those in the orbit queue perform sensing (at the

rate λ2 +θq2(t)), they find all channels occupied (with probability β(t)) but decide not to abandon

the system (with probability 1−φ); (2) When the unlicensed users finish a service session (at the

rate µz2(t)), they need another one (with probability 1− p) and find licensed users waiting (with

probability α(t)) but do not abandon the system (with probability 1−φ). The rate of decrease in

q2(t) is θq2(t), which is the rate the unlicensed users in queue sense for available channels.

13

When z1(t)+z2(t)< 1 or q1(t)> 0, the system dynamics is quite simple and resembles that of the

many-server queues in call center applications. For example, when z1(t) + z2(t)< 1, the differential

equations (4)–(7) reduce to q1(t)≡ 0 and

z′1(t) = λ1−µ1z1(t),

z′2(t) = λ2 + θq2(t)− pµz2(t),

q′2(t) =−θq2(t).

Otherwise, the system dynamics is more complicated. Moreover, the process x(·) can move back

and forth among different cases, which makes the analysis even more challenging as shown in

Appendix A.

Despite the complexity, the fluid model can be solved numerically. Furthermore, we can obtain

the steady state of the fluid model in Theorem 1 to approximate the steady state of the original

system. For example, β := limt→∞

β(t) and TH2 := limt→∞

pµz2(t) can be used to accurately approximate

the steady-state delay probability of the licensed users and the throughput rate of the unlicensed

users, respectively. Note that fluid models fail to yield probabilistic performance measures in most

applications. Similar to Gurvich and Perry (2012), our fluid model actually provides accurate

approximations for them.

4.2. The Steady State of the Fluid Model

While the offered load of such a system is λ1µ1

+ λ2µ2

, the effective load is endogenous as the average

time for which an unlicensed user occupies a channel 1µ

defined in (2) depends on the decision 1µt

.

Since 1p

is the average number of service sessions needed to fulfill the service requirement of an

unlicensed user, the effective service time of an unlicensed user is 1pµ

. Thus, the effective load of

the system isλ1

µ1

+λ2

pµ,

where pµ= µ2µsµt+µs

. Note that the effective load is always no less than the offered load and equals

the offered load if and only if there is no restriction on the service process of the unlicensed users,

i.e., 1µt

=∞. The shorter the transmission time in a service session, the more service sessions (and

hence sensing) are needed for the unlicensed users to complete their transmissions and the more

congested the system is. Depending on the effective load of the system, the steady state of the fluid

limits are given in the next theorem whose proof can be found in Appendix A.

Theorem 1. There exists a unique solution1 to the fluid model. Moreover, the limiting behavior

of the fluid model as t→∞ can be characterized as follows.

1 A vector-valued function x(t) is called a solution of the fluid model if it is absolutely continuous on every closedtime interval and satisfies equations (4)–(7) almost everywhere.

14

1. If λ1µ1

+ λ2pµ> 1, then lim

t→∞x(t) =

(λ1µ1,0,1− λ1

µ1, 1−φθφ

[λ2− pµ

(1− λ1

µ1

)]), TH2 = pµ

(1− λ1

µ1

), and

(β,α) is the unique solution to

α=λ1

λ1 +µ(1− λ1

µ1

)β, (10)

γ = β+ (1−β)(1− p)α

p+ (1− p)α, (11)

λ2E[γK ] = λ2− pµ(1− λ1

µ1

), (12)

where K ≥ 1 follows a geometric distribution with parameter φ.

2. If λ1µ1

+ λ2pµ≤ 1, then lim

t→∞x(t) =

(λ1µ1,0, λ2

pµ,0)

, TH2 = λ2, and α= β = 0.

We first describe the intuition behind the delay probability β in Equations (10)–(12) before

discussing the steady-state behavior in more detail in the next section. Equation (10) is obtained

by plugging limt→∞

x(t) into (9). Note that 1− β is also the probability that an unlicensed user will

be served upon arrival or after each sensing activity while waiting in the orbit queue, and (1−p)αp+(1−p)α

is the probability that an unlicensed user in service will be interrupted. Thus, γ in (11) is the

probability that an unlicensed user will experience blockage or interruption and hence needs to

decide whether or not to abandon the system at least once. Since K ≥ 1 represents the number of

times an unlicensed user needs to decide whether to abandon the system, E[γK ] is the probability

that an unlicensed user will abandon the system. So the left-hand side of (12) can be understood

as the abandonment rate of the unlicensed users, while the right-hand side is also the abandonment

rate but calculated by subtracting the rate pµ(

1− λ1µ1

)at which unlicensed users complete their

service from the total arrival rate λ2. Given that E[γK ] = γφ1−γ(1−φ)

, we actually have a closed-form

expression (see (29) in Appendix A) for the delay probability β from solving (10)–(12).

Table 1 also presents a comparison between the simulated delay probability and throughput rate

and the approximation based on the fluid model. As one can see, the fluid approximation works

well, especially when n is large, which is the case in our application. Furthermore, our simulation

also reveals that the average queue length of the licensed users is indeed quite short (vanishes

asymptotically). For instance, the 0.95 confidence interval of the queue length of the licensed users

is 0.0172± .0001 with deterministic times and 0.0211± .0001 with exponential times when n= 4000

and 1µt

= 0.6.

From Theorem 1, we can see that the system performance is insensitive to θ, the frequency

at which the unlicensed users sense for an available channel while waiting in the orbit queue.

This is because, although θ affects the transient of the differential equations (4)–(7), it influences

the steady state through the total sensing speed θq2 (i.e., when the derivatives of the left hand

side equal 0). As θ increases, the unlicensed users are allowed to sense channel availability more

15

frequently and hence abandon the system sooner, which lowers the number of waiting unlicensed

users q2. It turns out that, under such a mechanism, the total sensing speed remains constant as θ

varies. The insensitivity of θ on the system performance is further confirmed by a simulation study

in Appendix C. Thus, the decision on the sensing frequency should be based on technological (e.g.,

power consumption as sensing consumes power) rather than operational concerns.

When the system is effectively under or critically loaded, in which case the offered load is

λ1µ1

+ λ2µ2≤ 1, all users will be served without delay in the steady state and no restriction needs to

be imposed on the unlicensed users. When the system is effectively overloaded, in which case the

offered load may or may not be above 1, only pµ(

1− λ1µ1

)of the unlicensed users will finish service

per unit time and the unlicensed users will experience interference with a positive probability.

Theorem 1 also reveals some interesting steady-state behavior that differs from that of the fluid

models in most applications such as call centers.

1. It is well understood in the queueing literature that, if a system is critically loaded, there is

a positive probability that delay will occur, even with an extra capacity of O(√λn) in most

non-preemptive models in applications such as call centers (see Halfin and Whitt 1981). In our

application, the requirement for unlicensed users to sense channel availability while waiting in

the orbit queue guarantees the availability of idle channels for all licensed users upon arrival

even when the system is critically loaded, leading to a zero delay probability for licensed

users asymptotically. We note a similar result in Pang and Perry (2015) that, by controlling

when outbound calls can be made, reserving a logarithmic order number of servers in a call

center can achieve a zero delay probability for inbound calls asymptotically when the system

is critically loaded.

2. It is also well understood that, when a system is overloaded, customers will experience delay

almost surely in most call center applications because all servers are busy all the time (see

Whitt 2006). In our application, however, an arriving licensed user still has a chance to enter

service upon arrival even when there is a large number of unlicensed users in the orbit queue

as it takes time for them to sense channel availability. Hence, the delay probability of the

licensed users, which is endogenously determined by the load through (10)–(12), is strictly

less than 1. Even if a licensed user is delayed upon arrival, his waiting time is in the order of

O(

1λn1

), which is relatively short but may still be significant in data transmission.

Essentially, the restriction that the unlicensed users are not allowed to sense channel availability

constantly makes the system operate more like a preemptive one for the licensed users than a

non-preemptive one.

16

4.3. Sensitivity of the System Performance

By Theorem 1, sensing frequency does not affect the system performance. Thus, we will focus on

the impact of the length of transmission time 1µt

(or equivalently the length of the service session)

on the throughput of the unlicensed users TH2 and the delay probability of the licensed users β.

Corollary 1. TH2 is always increasing in the transmission time 1µt

, i.e., allowing the unlicensed

users longer service sessions will increase the system throughput rate. β is quasi-convex in 1µt

. More

specifically,

• If λ1µ1

+ λ2µ2≤ 1 or 1

µs≥

1−µ2λ2

(1−λ1µ1

)1+

µ2λ1

(1−λ1µ1

) 1µ2

, then β decreases in 1µt

(see Figure 2(a)–(b)).

• Otherwise, there exists a threshold 1µt<∞ such that β decreases in 1

µtwhen 1

µt≤ 1

µtand

increases in 1µt

when 1µt> 1

µt(see Figure 2(c)–(d)).

Figure 2 illustrates the delay probability as a function of the transmission time for various λ1, λ2

and 1µs

when 1µ1

= 1µ2

= 1, θ= 0.4 and φ= 0.5. Note that the purpose of restricting the amount of

time the unlicensed users can occupy a channel is to limit the interference of the unlicensed users

to the service of the licensed users. Thus, intuitively, shorter service sessions should always lead

to a lower delay probability. The corollary reveals that this is true only if 1µt

= 0 which happens

when the workload from both types of users are high enough and the sensing time is moderate

(Figure 2(d)). When the system is overloaded and sensing is not too time consuming, imposing

too short service sessions will only increase the effective load and hence the delay probability,

while imposing relatively longer service sessions will increase the delay probability as expected

(Figure 2(c)). When the system is under or critically loaded, shorter service sessions will either

have no impact on the delay probability or turn the system into an effectively overloaded one,

increasing the delay probability (Figure 2(a)). When the system is overloaded and sensing takes a

long time, it only makes sense to allow an unlicensed user to transmit for a significant amount of

time in order to lower the delay probability (Figure 2(b)).

4.4. Optimal Sharing Decisions in the Steady State

In this section, we investigate whether a given band of spectrum should be shared with unlicensed

users and the transmission time 1µt

that maximizes the throughput rate of the unlicensed users

while keeping the delay probability of the licensed users below a certain level, η. Note that θ does

not affect the system performance by Theorem 1 and the transmission time is the only decision.

Furthermore, maximizing the throughput rate of the unlicensed users is equivalent to maximizing

the throughput rate of the system since the throughput rate of the licensed users is a constant.

17

0.00 0.05 0.10 0.15 0.201µt

0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

β

λ2 =0.79

λ2 =0.8

effectively over loaded effectively under loaded

effectively critically loaded

(a) Under (λ1 = 0.2, λ2 = 0.79) and critically (λ1 = 0.2,

λ2 = 0.8) loaded, 1µs

= 0.001

0.0 0.1 0.2 0.3 0.4 0.5 0.61µt

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

β

(b) Overloaded (λ1 = 0.2, λ2 = 0.9) and 1µs

= 0.01 >

0.022

0.0 0.1 0.2 0.3 0.4 0.5 0.61µt

0.17

0.18

0.19

0.20

0.21

0.22

0.23

β

(c) Overloaded (λ1 = 0.2, λ2 = 0.9) and 1µs

= 0.0001<

0.022

0.0 0.2 0.4 0.6 0.8 1.01µt

0.890

0.895

0.900

0.905

0.910

0.915

0.920

β

(d) Overloaded (λ1 = 0.8, λ2 = 1.7) and 1µs

= 0.25 ∈

[0.23,0.4]

Figure 2 The delay probability as a function of the transmission time

4.4.1. Whether and How to Share When the system is under or critically loaded, the

system may also be effectively overloaded if one allocates shorter service sessions to the unlicensed

users. However, by Theorem 1, even if the unlicensed users are allowed to transmit for as long

as they need, the delay probability converges to zero and all users are able to complete their

transmission without delay as n→∞. Thus, we do not need to restrict the service session of the

unlicensed users when n is large enough.

When the system is overloaded, it is also effectively overloaded regardless of the length of the

allocated service session. By Theorem 1, TH2 = pµ(

1− λ1µ1

)and β is the solution to (12). Thus,

the optimization problem can be written as

maxµt≥0

pµ (13)

18

s.t. β ≤ η,

µ=(µ2 +µt)µsµt +µs

,

p=µ2

µ2 +µt.

Since the objective function is increasing in 1µt

, the optimization problem reduces to one of finding

the largest 1µt

that satisfies the delay constraint. When η is so small that the feasible region is

empty, no unlicensed users should be allowed in the system. Once η is large enough to make

the feasible region non-empty, unlicensed users will be allowed in the system. As η increases, the

optimal transmission time 1µ∗t

increases. The optimal transmission time 1µ∗t

=∞, i.e., the unlicensed

users are allowed to complete their transmission once they start occupying a channel, if η is larger

than the point such that the feasible region becomes unbounded.

Figure 3 demonstrates the optimal spectrum sharing decision as a function of η and λ2 when

λ1 = 0.2, 1µ1

= 1µ2

= 1, 1µs

= 0.001, θ = 0.4 and φ= 0.5. The upper curve specifies the arrival rate

above which the unlicensed users should not be allowed to share the spectrum and the lower one

is the threshold below which there is no need to restrict the service session of the unlicensed users,

i.e., 1µ∗t

=∞.

Since our analysis only holds in an asymptotic sense (as the number of channels n becomes large),

there is still a non-negligible delay probability when the system is under or critically loaded and

n is not sufficiently large. For the same example in Figure 3 withλn1n

= 0.2, Figure 4 demonstrates

the optimal sharing decisions, obtained through simulation, as a function of η andλn2n

for n =

100,200,500,1000, in which case the system is under or critically loaded whenλn2n≤ 0.8. As one can

see, the structure of the optimal sharing decision remains the same, and as n increases, sharing is

more likely to occur and the unlicensed users should be allowed longer service sessions.

4.4.2. Sensitivity of the Optimal Decision The optimal decision 1µ∗t

and the throughput

rate of the unlicensed users TH∗2 depend on the system parameters in the following way.

Proposition 1. The optimal 1µ∗t

decreases, i.e., the unlicensed users are allowed a shorter trans-

mission time, as

(1) λ1 increases while keeping λ1 +λ2 constant when µ1 = µ2;

(2) λ1 and µ1 decrease while keeping λ1µ1

constant; and

(3) λ2 and µ2 increase while keeping λ2µ2

constant.

Furthermore, the optimal throughput TH∗2 will decrease under (1) and (2).

Note that under all the scenarios, the total offered load λ1µ1

+ λ2µ2

is kept constant. Proposition 1

states that shorter service sessions should be allocated to the unlicensed users (1) as the proportion

19

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14η

0.80

0.82

0.84

0.86

0.88

λ2

do not share

share with restriction

share without restriction

Figure 3 The optimal sharing decision as a function of η and λ2 for an overloaded system

of licensed users increases when all users have identical service requirements, (2) if there are fewer

licensed users but with longer service times; and (3) if there are more unlicensed users but with

shorter service times. While (1) and (3) are more intuitive, (2) holds because the delay probability

only depends on both λ1µ1

and λ1. A delay incident of a licensed user is counted as one regardless

of his service requirement. With fewer licensed users, each delay contributes more to the delay

probability and it is easy to show that shorter service sessions should be imposed on the unlicensed

users.

As a result, the optimal throughput rate pµ∗(

1− λ1µ1

)= µ2µs

µ∗t+µs

(1− λ1

µ1

)decreases under scenarios

(1) and (2) as expected. These suggest that spectrum sharing is beneficial to systems with a smaller

proportion of licensed users or a large number of licensed users with shorter service times. Under

scenario (3), although shorter service sessions have a negative impact on the throughput rate due

to more sensing activities, the increase in the number of unlicensed users with shorter service times

has a positive impact. Thus, the impact on throughput rate is not monotone.

20

0.02 0.04 0.06 0.08 0.10.7

0.75

0.8

0.85

do not share

share

withres

tricti

on

share without restriction

η

λn2n

(a) n= 100

0.02 0.04 0.06 0.08 0.10.7

0.75

0.8

0.85

do not share

share withres

triction

share without restriction

η

λn2n

(b) n= 200

0.02 0.04 0.06 0.08 0.10.7

0.75

0.8

0.85

do not share

share withrestri

ction

share without restriction

η

λn2n

(c) n= 500

0.02 0.04 0.06 0.08 0.10.7

0.75

0.8

0.85

do not share

share with restriction

share without restriction

η

λn2n

(d) n= 1000

Figure 4 The optimal sharing decisions as a function of η andλn2n

5. Justifications for the Fluid Approximation

In this section, we demonstrate in Theorem 2 that the scaled process Xn(t) converges to the fluid

model x(t) in Section 4. Since the proof of the theorem is quite involved, we describe the main

ideas of the proof through the construction of the fluid model, especially the instantaneous delay

probability of the licensed users β(t) and the instantaneous interruption probability of the unli-

censed users α(t). The complete proof can be found in Appendix B. For any T > 0, let D([0, T ],R4)

be the space of all right-continuous R4 valued functions on [0, T ] with left limits, endowed with

the Skorohod J1 topology. Let “⇒” denote convergence in distribution for random objects in R4

equipped with Euclidian topology or D([0, T ],R4) with Skorohod J1 topology.

Theorem 2 (Fluid Approximation). Under the asymptotic regime, if Xn(0)⇒ x(0) as n→∞,

then Xn(t)⇒ x(t) in D([0, T ],R4), where x(t) is the fluid model specified in Definition 2.

Need for both scaled and unscaled processes If we let Λni (t) denote the Poisson process

with the rate λni , then Λn1 (t+ δ)−Λn

1 (t) is the total number of licensed users arriving in a small

21

interval [t, t+ δ], among which∫ t+δt

1{In(s−)=0}dΛn1 (s) will find no idle channels upon arrival and

have to wait. Thus, the delay probability of the licensed users during this small time interval

is E[ ∫ t+δ

t 1{In(s−)=0}dΛn1 (s)

Λn1 (t+δ)−Λn1 (t)

]. That is, the delay probability depends on the information about the

unscaled process In(t)≥ 0 as it determines whether an unlicensed user in service should vacate a

channel at the end of a service session. Likewise, we need to keep track of the unscaled process of

the queue length of the licensed users Qn1 (t) and obtain the probability of an unlicensed user in

service being interrupted in [t, t+ δ]. However, In(t)≥ 0 vanishes in the asymptotic regime along

with the process Qn1 (t)≥ 0 as in most systems with multiple classes and we need to keep track of

both the scaled and unscaled processes in order to obtain the system dynamics and asymptotic

system performance.

The system dynamics using the averaging principle To obtain the system dynamics, we

need to apply the averaging principle by first expressing the probabilities in [t, t+ δ] as a time

average using PASTA (Poisson arrivals see time average). For instance, the fraction of time for

which there is no idle channel in the system is

1

δ

∫ t+δ

t

1{In(s−)=0}ds=1

nδ

∫ t+nδ

t

1{In(t+ s−n )=0}ds. (14)

Let

mn(t) =Qn1 (t)− In(t). (15)

We study the system dynamics for the unscaled process mn(t+ s

n

)for 0≤ s≤ nδ. Note that the

process mn(t+ ·

n

)oscillates around zero in the order of 1. When mn

(t+ s

n

)< 0 (there are idle

channels and no licensed users waiting by (1)), the process increases by 1 when there is a new arrival

at the rate λn1 + λn2 or one of the unlicensed users in the orbit queue enters service after sensing

the system at the rate θQn2 (t+ s

n). The process decreases by 1 when a user (licensed or unlicensed)

completes service at the rate µ1Zn1 (t+ s

n) + pµZn2 (t+ s

n). When mn

(t+ s

n

)> 0 (there are licensed

users waiting and no idle channels), the process increases by 1 at the rate λn1 and decreases at the

rate µ1Zn1 (t+ s

n)+µZn2 (t+ s

n). We refer readers to Appendix B.1 for the detailed system dynamics.

It is the long-run average behavior of mn(t+ ·

n

)that plays the key role in determining the fraction

in (14) when n becomes large in the asymptotic regime.

Explanation for β(t) and α(t) As one can see, the process mn(t+ ·

n

)is not a Markov process

since its evolution depends on a higher dimension process than itself. However, if we approximate

the above mentioned rates by their fluid counterparts, i.e., Zni (t+ sn

) by zi(t), Qn2 (t+ s

n) by q2(t),

and λni by λi, we have a Markov process as in Figure 5 whose steady-state distribution πt can be

easily obtained. We use πt(j) to approximate the asymptotic proportion of time for which there

are j licensed users in the queue when j > 0 and there are −j idle channels when j < 0. The delay

22

0−1−2· · · 1 2 · · ·

µ1z1(t) + pµz2(t)

λ1 +λ2 + θq2(t)

µ1z1(t) + pµz2(t)

λ1 +λ2 + θq2(t)

µ1z1(t) + pµz2(t)

λ1

µ1z1(t) +µz2(t)

λ1

µ1z1(t) +µz2(t)

λ1

Figure 5 The asymptotic transition diagram of mn(t+ ·

n

).

probability of the licensed users in (14) and the interruption probability of the unlicensed users in

the asymptotic regime can be approximated by∞∑j=0

πt(j) := β(t) and∞∑j=1

πt(j) := α(t), respectively.

6. Extensions

In this section, we extend the problem to systems with time-varying arrival rates and propose a

diffusion approximation that can lead to better performance in some cases.

6.1. With Time-Varying Arrival Rates

When the arrival rates vary over time, the optimal decision on the transmission time needs to be

adjusted dynamically. Suppose that adjustment of the transmission time can be done instanta-

neously and the initial state x(0) is given. We can extend the fluid model in Definition 2 to allow

time-varying arrivals by adding an argument t to λi, µt, p, and µ to denote their instantaneous val-

ues. Following similar arguments in Appendices A and B, we can show that the stochastic processes

with time-varying arrival rates converge to the extended fluid model and there exists a unique

solution to time-varying differential equations of the fluid model as long as λi(t)’s are bounded and

locally Lipschitz continuous. In this case, the instantaneous throughput rate is p(t)µ(t)z2(t), the

instantaneous delay probability β(t) is given by (8) and the optimization problem over a period of

time T can then be written as

maxµt(·)

∫ T

0

p(t)µ(t)z2(t)dt

s.t. β(t)≤ η,

µ(t) =[µ2 +µt(t)]µsµt(t) +µs

,

p(t) =µ2

µ2 +µt(t).

Although such a continuous-time dynamic programming problem can be solved numerically using

policy iteration, the resulting policy is hard to implement in practice. Thus, we ask whether a

periodically adjusted policy will work well. As an example, suppose that the arrival rates change

over time as in Figure 6, 1µ1

= 1µ2

= 1 minutes, 1µs

= 0.02 minute, θ = 0.4, φ = 0.5 and η = 0.2.

Figure 6 plots the transmission times adjusted on an hourly basis and the performance over a

10-hour period. As one can see, our heuristic policy performs very well. According to our numerical

23

experiments, the throughput rate under the hourly adjusted policy is consistently within 0.2% of

the optimal throughput rate.

0.00.20.40.60.81.0

λi(t)

λ1 (t)λ2 (t)

0.00.20.40.60.81.0

TH

2(t

)

optimal TH ∗2 (t)

heuristic TH2 (t)

0.000.050.100.150.200.25

β(t

)

optimal β ∗ (t)

heuristic β(t)

0 2 4 6 8 10Time (hour)

0.0

0.5

1.0

1.5

2.0

µt(t

)

optimal µ ∗t (t)

heuristic µt(t)

Figure 6 With time-varying arrival rates and periodic adjustments

6.2. A Diffusion Scaling

Although our fluid scaling results in good approximations, it leads to a zero delay probability when

the system is under and critically loaded, which is not accurate when n is small. Thus, we ask

whether a diffusion scaling may work better for under and critically loaded systems.

Consider the diffusion scaling where the licensed users grow in the order of O(√n) and the

unlicensed in O(n), i.e.,

λn1 = λ1

√n,

λn2 = pµn+ λ2

√n,

and Zn1 (t) =Zn1 (t)√

n, Zn2 (t) =

Zn2 (t)−n√n

and Qni (t) =

Qni (t)√n

are the corresponding diffusion scaled pro-

cesses. Such a scaling explicitly assumes that there are far more unlicensed users than licensed

24

users. It can be shown that the diffusion scaled processes converge and there exist coefficients C1,

C2, Cq, Cβ and Cα such that

E[Zn1 (∞)] =C1

√n+ o(

√n), E[Zn2 (∞)] = n+C2

√n+ o(

√n),

E[Qn1 (∞)] = o(1), E[Qn

2 (∞)] =Cq√n+ o(

√n),

αn =Cα√n

+ o( 1√

n

), βn =

Cβ√n

+ o( 1√

n

). (16)

6.2.1. Estimation of the coefficients First, it is easy to see that C1 = λ1µ1

since the licensed

users do not abandon. Since the unlicensed users may abandon, the system is stable in the long run

and hence the balance equations are given by letting (4)–(7) to be zero and replacing (λi, zi, qi, β,α)

by(λni ,E[Zni (∞)],E[Qn

i (∞)], βn, αn). Solving the balance equations, we are able to obtain

Cα = 0,

Cβ =θCq

(1−φ)pµ, (17)

θCq = (1−φ)[λ2− pµC2 + θCq

]. (18)

It remains to estimate C2 and Cq. If we are able to derive a closed-form steady state distribution

of the limit of the diffusion scaled process, we can obtain the value of these coefficients. Although the

four-dimensional diffusion scaled process, (Zn1 , Qn1 , Z

n2 , Q

n2 ), can be reduced to a three-dimensional

process as Qn1 converges to 0, it has some complicated reflection behavior on the boundary when

all channels are busy, i.e., Zn1 (t) + Zn2 (t) = 0. In general, it is challenging to derive the steady state

distribution of a multidimensional diffusion process and closed-form expressions of the coefficients

are almost impossible. Thus, we propose a heuristic method to derive closed-form approximations

for C2 and Cq and hence Cβ.

We pretend that the licensed users occupy λ1µ1

√n channels exclusively and the waiting unlicensed

users form a steady source of arrival with the rate θCq√n. The unlicensed users are served by

the remaining n− λ1µ1

√n channels and forms an Erlang-B queue with the arrival rate λn2 + θCq

√n

and service rate pµ. In such a network, limn→∞

Zn2 (t) = z2(t) is a reflected Brownian motion with

an infinitesimal mean −pµ(z2− λ2+θCq

pµ

)and infinitesimal variance 2pµ. Therefore, lim

t→∞limn→∞

Zn2 (t)

follows a truncated normal distribution with meanλ2+θCqpµ

and variance 1 on(−∞,− λ1

µ1

)and hence

C2 =E[

limt→∞

limn→∞

Zn2 (t)]

=λ2 + θCq

pµ−

Φ′(− λ1µ1− λ2+θCq

pµ

)Φ(− λ1µ1− λ2+θCq

pµ

) , (19)

where Φ(·) is the cumulative distribution function of the standard normal distribution. By (17)–

(19), we can obtain C2, Cq, and

Cβ =Φ′(− λ1µ1− λ2

pµ− (1−φ)Cβ

)Φ(− λ1µ1− λ2

pµ− (1−φ)Cβ

) . (20)

25

By (16), the delay probability of the nth system can be approximated byCβ√n

, the throughput rate

can be approximated by pµE[Zn2 (∞)/n] = pµ(1 +C2/√n). Thus, the accuracy of the estimation of

the system performance is reflected by the coefficients Cβ and C2.

6.2.2. Accuracy of the heuristic To show how the above heuristic approximates the delay

probability and throughput rate of the diffusion scaled processes as well as the actual system,

we conduct a numerical experiment. We simulate large systems to obtain the diffusion limits and

compare them with the heuristic ones. For 1µ1

= 1µ2

= 1, 1µs

= 0.0001, θ = 0.4, φ = 0.5 and 1µt

=

∞, Figures 7 and 8 compare the diffusion scaled delay probabilities with the heuristic ones. As

expected, our heuristic mimics the performance of the simulated diffusion limits well, especially

when the systems are under or critically loaded. Note that, in the network, all the channels are

pooled to serve both licensed and unlicensed users, while the heuristic estimates the coefficients

pretending that the licensed users occupy a fixed number of channels. When the system is under

or critically loaded, the heuristic works well as long as the channels are well allocated to the two

types of users, as shown in Figure 7(a) and Figure 8. The impact of decoupling the channels is

higher when the system is over loaded, as shown in Figure 7(b).

0 500 1,000 1,500 2,000 2,500

0

0.02

0.04

0.06

0.08

√n

βn

The actual system

Simulated diffusion

Heuristic

Fluid

(a) under loaded: λ1 = 0.05 and λ2 = 0.75

0 500 1,000 1,500 2,000 2,500

0.3

0.35

0.4

0.45

0.5

0.55

√n

βn

The actual system

Simulated diffusion

Heuristic

Fluid

(b) over loaded: λ1 = 0.2 and λ2 = 1.0

Figure 7 Under and over load: the delay probabilities as a function of n when λn1 = λ1n and λn2 = λ2n

6.2.3. Comparison to the Fluid Approximation Figures 7 and 8 also plot the delay

probabilities under the fluid scaled processes and of the actual systems. As one can see, the fluid

approximation always underestimates β, while the diffusion approximation always overestimates

it. Furthermore, the fluid approximation outperforms the diffusion approximation when the system

26

0 10 20 30 40 50

0

0.1

0.2

√n

βn

The actual system

Simulated diffusion

Heuristic

Fluid

(a) λ1 = 0.5 and λ2 =−0.5

0 10 20 30 40 50

0

0.1

0.2

0.3

0.4

√n

βn

The actual system

Simulated diffusion

Heuristic

Fluid

(b) λ1 = 1.0 and λ2 =−0.5

Figure 8 Critical load: the delay probabilities as a function of√n when λn1 = λ1

√n and λn2 = pµn+ λ2

√n

is over loaded, and vice versa when the system is under or critically loaded. Thus, neither method

is uniformly more accurate than the other. However, further comparisons reveal the following.

1. The fluid scaling leads to analytical closed-form approximations, while analysis under the

diffusion scaling involves solving the steady states of multi-dimensional diffusion processes,

which is known to be an open question in most cases.

2. The closed-form approximations under the fluid scaling reveal important operational insights

(in Section 4) that are not obvious under the diffusion approximation.

3. The diffusion approximation may not be feasible when a system is overloaded or the num-

ber of licensed users is comparable to or more than that of unlicensed ones, while the fluid

approximation can be used under any load level with any ratio between the licensed and unli-

censed users. Such a drawback may further limit the diffusion approximation to be adopted

to systems with time varying or random arrivals.

7. Conclusions and Future Research

Opportunistic access of licensed spectrum by unlicensed users is widely considered as a way to

alleviate artificial scarcity of radio spectrum by increasing the spectrum utilization. However, it

may reduce the service quality for licensed users due to potential interference from unlicensed

users. While much research on spectrum sharing has been conducted by researchers in electrical

engineering with the main focus on the technological issues, the operational aspects have not been

adequately addressed through analytical work.

In this paper, we model a shared network consisting of both licensed users and unlicensed users

as a multi-class, many-server queueing system. The distinctive features of our model are that the

27

service requirement of an unlicensed user can be fulfilled even after multiple interruptions and

the unlicensed users waiting in the queue are required to sense channel availability periodically

while waiting. These features complicate system dynamics and lead to quite different insights from

those derived from most service systems. We show that the sensing frequency of the unlicensed

users waiting in the queue does not affect system performance from the operational perspective

and its decision should be based on technological concerns. When the system is under or critically

loaded, there is no need to restrict the service session of unlicensed users. Otherwise, limiting the

transmission of the unlicensed users is necessary only when the system load is above a threshold.

Thus, it is possible to improve spectrum utilization while guaranteeing a very high service level,

expected by licensed users in practice, and spectrum sharing can potentially be a socially optimal

solution to alleviating spectrum scarcity.

Spectrum sharing, if feasible, is especially beneficial for systems with a smaller portion or a large

number of licensed users with shorter service times. Our study sheds light on the implementation

of spectrum sharing and opens the door for new applications of existing queueing theory in wireless

communication networks, which may lead to the development of new methodologies.

Our study also provides some rich research opportunities. For instance, the arrival rates of the

users may be uncertain in practice. Our preliminary result shows that higher variance will always

hurt system performance if the system is expected to be under or critically loaded. However, if

the system is expected to be overloaded, it seems that increasing the variability up to a certain

level will actually improve the throughput rate. Thus, research needs to be done to investigate the

impact of uncertain arrival rates on system performance.

In reality, users’ behavior in data transmission can be more complicated than those in the network

in Figure 1. For instance, unlicensed users who have to abandon the system earlier may reenter

the system later, while licensed users may abandon the system if no idle channel is available upon

arrival. Also, sensing may not be perfect, e.g., a false alarm can occur, in which case a spectrum

opportunity is overlooked by an unlicensed user. It will be interesting to incorporate these elements

into the model and examine how they change the system performance and operational decisions.

The insights revealed in the above research may also pave the way for studying other important

business issues in wireless communications such as contract design and pricing in shared networks.

For instance, how should a spectrum owner set the prices and decide the service quality to both

licensed and unlicensed users in a shared network? Should unlicensed users be charged a fixed

and/or usage based fee? Since unlicensed users may belong to different service providers, should a

spectrum owner run an auction to select the service providers and settle the prices?

Acknowledgments

28

The authors acknowledge the advice of Professor Qian Zhang from the Department of Computer Science

at Hong Kong University of Science and Technology, and encouragement of the area editor, the associate

editor, and the two anonymous reviewers. This work was supported by the Hong Kong Research Grants

Council under Grants 622713, 16500615, and 16501015, and Hong Kong Polytechnic University under Grants

252019/16E and 1-ZE5G.

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Brief Bios of Authors

Shining Wu is an assistant professor in Logistics and Maritime Studies at the Hong Kong Poly-

technic University. His research interests include supply chain management, strategic consumer

behavior, queueing theory and its applications, operational issues in spectrum sharing, and data-

driven optimization for queueing systems.

Jiheng Zhang is an associate professor in Industrial Engineering and Logistics Management at the

Hong Kong University of Science and Technology. His research interests are in applied probability,

stochastic modeling and optimization, data analysis, numerical methods and algorithms.

Rachel Q. Zhang is a professor in Industrial Engineering and Logistics Management at the

Hong Kong University of Science and Technology. Her research interests include supply chain and

inventory management, stochastic analysis of service operations, and the interface of finance and

operations. Her research has appeared in such journals as Operations Research, Management Sci-

ence, Interfaces, IIE Transactions, Naval Research Logistics, and Advances in Applied Probability.